Estimating Taylor Series
C
id
T
l
l
i l
f
f
ti
f( )
Consider a Taylor polynomial of a function f(x):
)
(
,
!
)
)(
(
)
(
n
n
a
f
n
a
x
a
f
x
T
Then we find that the approximation stopping after n = k will be given by:
(somewhat similar to that of the alternating series test)
0
n
)!
1
(
)
)(
(
)
(
1
)
1
(
k
a
x
z
f
x
R
k
k
k
Where z is some (not usually known) number between a and x depending on how
d
i
ti
f
d I
h
t
th t R ( ) i th
i d
f
many derivatives were performed. In here, we note that R
k
(x) is the remainder of
how close f(x) is to the Taylor expansion stopping after k terms.
Proving this theorem is a bit challenging, so we will not do so, but if we can place a
reasonable bound on z, we can show that the error is

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Estimating Taylor Series
S
h
d
f( )
T( )? C
id
th
f ll
i
Th
So when does f(x) = T(x)? Consider the following Theorem:
Consider a Taylor Series of a function f(x):
0
)
(
,
!
)
)(
(
)
(
n
n
n
a
f
n
a
x
a
f
x
T
Then T(x) = f(x) if and only if T(x) – f(x) = 0, if and only if
Then T(x)
f(x) if and only if T(x)
f(x)
0, if and only if
0
)
(
)
(
lim
x
T
x
f
k
k
0
)
(
lim
x
R
k
k
0
)!
(
)
)(
(
lim
1
)
1
(
k
a
x
z
f
k
k
1
k